One well known proof of the Dirichlet Integral is to use the kernel $\frac{\sin{(2n+1)x}}{\sin{x}}$ and applying the Riemann-Lebesgue Lemma to an auxiliary function to obtain an equivalence between two integrals.
There exists a similar proof for the kernel $\frac{\sin^2{nx}}{\sin^2{x}}$ but does not require the Riemann-Lebesgue Lemma as it is possible to directly bound the difference and show it is $\mathcal{O}(1/n)$.
In either case, the proof implicitly assumes the limit of a subset or subsequence is the limit of the entire set/sequence.
The point is, the careful reader who is not exposed to higher calculus and non-elementary techniques may choose to ask, what if I reject the choice to take this limit over this particular countable set? How does this prove the limit* actually exists and is not just an artefact of the choice of the countable set the limit is taken over? What if the limit* is taken over the entire set of reals and fails to exist? (Obviously it does exist, but the student doesn't know that, and the above "proof" doesn't show the limit* exists, only it exists over a particular subset of $\mathbb{R}$.)
I would like to know if there is a low-tech/elementary means of resolving this conundrum.
Alternatively, I would be happy to see an elementary proof of the Dirichlet Integral, if any watertight and airtight proof exists.
* "the limit" refers to the following: $\displaystyle \lim_{n \to \infty} \int_{-(2n+1)\pi}^{(2n+1)\pi} \frac{\sin{x}}{x} \, \mathrm{d}x$
and to the careful outside observer, it is not obvious that this proves the limit of the sinc integral exists.